Abstract

Distinguishing between direct and indirect connections is essential when interpreting network structures in terms of dynamical interactions and stability. When constructing networks from climate data the nodes are usually defined on a spatial grid. The edges are usually derived from a bivariate dependency measure, such as Pearson correlation coefficients or mutual information. Thus, the edges indistinguishably represent direct and indirect dependencies. Interpreting climate data fields as realizations of Gaussian Random Fields (GRFs), we have constructed networks according to the Gaussian Graphical Model (GGM) approach. In contrast to the widely used method, the edges of GGM networks are based on partial correlations denoting direct dependencies. Furthermore, GRFs can be represented not only on points in space, but also by expansion coefficients of orthogonal basis functions, such as spherical harmonics. This leads to a modified definition of network nodes and edges in spectral space, which is motivated from an atmospheric dynamics perspective. We construct and analyzenetworks from climate data in grid point space as well as in spectral space, and derive the edges from both Pearson and partial correlations.Network characteristics, such as mean degree, average shortest path length, and clustering coefficient, reveal that the networks posses an ordered and strongly locally interconnected structure rather than small-world properties. Despite this, the network structures differ strongly depending on the construction method. Straightforward approaches to infer networks from climate data while not regarding any physical processes may contain too strong simplifications to describe the dynamics of the climate system appropriately.

Received 22 October 2013Accepted 21 March 2014Published online 04 April 2014

Lead Paragraph: Complex networks are increasingly used in climate sciences. However, the climate system does not consist of a structure with identifiable nodes. The assumption in previous studies is that the systems dynamics is described by the dynamics of oscillators, where each oscillator is associated with a grid point in space or a node. The edges are derived by thresholding a bivariate dependency measure, for instance Pearson correlation coefficients (PCCs), calculated from the time series of an atmospheric state variable at the different nodes. The resulting networks map known patterns of the atmospheric circulation. However, interpreting these findings in terms of an interaction structure is problematic since the edges indistinguishably represent both direct and indirect dependencies. When we interpret climate data as realizations of Gaussian Random Fields (GRFs), the direct interactions between the nodes are recorded in the inverse covariance matrix, also called precision matrix. According to the Gaussian Graphical Model (GGM) approach, we construct networks with edges derived from partial correlations computed from the precision matrix. Furthermore, from an atmospheric dynamics perspective, the feasibility of describing the atmosphere as a spatial grid of oscillators is questionable. The coupled oscillator representation seems to be more valid in the space of spectral basis functions, such as the spherical harmonics. We have constructed networks in grid point space and spectral space, inferring edges from Pearson and from partial correlation coefficients. We show that the network characteristics are strongly dependent on the definition of nodes and edges. Seemingly straightforward approaches to infer network structures from climate data, independent of physical processes, may be insufficient to describe and interpret the dynamics of the climate system.

Acknowledgments:

The authors would like to thank Stephan Bialonski and the anonymous reviewers for their helpful comments.

Abstract

Distinguishing between direct and indirect connections is essential when interpreting network structures in terms of dynamical interactions and stability. When constructing networks from climate data the nodes are usually defined on a spatial grid. The edges are usually derived from a bivariate dependency measure, such as Pearson correlation coefficients or mutual information. Thus, the edges indistinguishably represent direct and indirect dependencies. Interpreting climate data fields as realizations of Gaussian Random Fields (GRFs), we have constructed networks according to the Gaussian Graphical Model (GGM) approach. In contrast to the widely used method, the edges of GGM networks are based on partial correlations denoting direct dependencies. Furthermore, GRFs can be represented not only on points in space, but also by expansion coefficients of orthogonal basis functions, such as spherical harmonics. This leads to a modified definition of network nodes and edges in spectral space, which is motivated from an atmospheric dynamics perspective. We construct and analyzenetworks from climate data in grid point space as well as in spectral space, and derive the edges from both Pearson and partial correlations.Network characteristics, such as mean degree, average shortest path length, and clustering coefficient, reveal that the networks posses an ordered and strongly locally interconnected structure rather than small-world properties. Despite this, the network structures differ strongly depending on the construction method. Straightforward approaches to infer networks from climate data while not regarding any physical processes may contain too strong simplifications to describe the dynamics of the climate system appropriately.